Simple Annuities

What are Annuities?

• An annuity is any sequence of payments that are made at equal (or fixed) periods of time.

• It can also be described as a continuous stream of equal payments made by one party to another over a specific period of time.

• Annuities can be classified into different types:

  • according to payment interval and compounding period
    • Simple: the payment interval is the same as the interval period
    • General: the payment interval and the compounding frequency are not the same.
  • according to time of payment
    • Ordinary Annuity: aka annuity immediate; payments are made at the end of each interval.
    • Annuity Due: payments are made at the start of each interval.
  • according to duration of annuity
    • Annuity Certain: annuities which has a definite start and end times.
    • Contingent Annuities: annuities in which the end time is not defined
      • The payments may be made for an indefinite amount of time.

• In this lesson, we will mainly focus on the concept of the Simple Annuity and how is it calculated.

Terms related to Annuity Computations

• We first want to get familiar to some terms we use for computing annuities.
  • When talking about annuities, it is often crucial to talk about the periodic interest rate instead of the annual interest rate.
  • The periodic interest rate can be computed as follows:

Formula
Periodic Interest Rate

• where:

• An another useful term is thinking about the total number of payments throughout the term of the annuity.
  • In such cases, we can use the formula below:

Formula
Number of Annuity Payments

• where:

• These formulas are often used for the sake of ease of computation.
  • We can begin practicing working through these formulas with a worked example.

Worked Example
Finding the Periodic Interest Rate and the Number of Payment Periods

Abigail signed a contract with a trusted lender where she pays them twice a month in exchange for her money to grow at annual interest rate compounding quarterly. If the contract lasts for years, find the periodic interest rate and the number of payments she will have to make.

Solution

• The annuity term is 5 years.
• The annual interest rate is or .
• The compounding frequency indicates that her money grows every quarter.
  • This means that it grows times a year.
• The payment frequency indicates that she pays the lender twice a month.
  • Since there are months every year, then she pays them times a year.

Now that we know what is known, we can now find the periodic interest rate and the number of payments.
Applying the formula:

Therefore, Abigail’s money grows at every quarter, and she has to make total payments during the length of the annuity.

The Future Value of Annuities

• To get a sense of the future value of an annuity is like, let us consider a scenario.

• Suppose that for each month, you invest 10010%35$ months?

  • For the first month, you invest 100$.
  • For the second month, the initial 10010%$.
    • Therefore, 100 \times 10% = $110$.
    • However, you also have to invest an another 100$.
  • For the third month, the 11010%$121$
    • At the same time, your 10010%$110$,
    • In addition, you also have to make an another 100$ investment.

• We can see that each investment that we make grows each compounding period.

  • To get their future value, we simply had to add up every investment we made plus the compounding interest it accumulated.
  • Therefore, we have:

• This is exactly the essence of what a future value of a simple annuity means!
  • For each investment we make, it independently grows by some interest rate.
  • The future value of the annuity simply tells us the total sum of all investments, plus the accumulated interest of each investment throughout the term of the annuity!

Definition
Future Value of an Annuity

The future value of an annuity is the sum of all annuity payments plus each of their accumulated interest throughout the rest of the term.

• In calculating annuities, we often had to deal annuities with a large number of payments.
• Fortunately for us, we don’t have to do a huge number of interest calculations for each investment.
  • Instead, we can simply use this formula:

Formula
Future Value of a Simple Annuity

• where:

• The proof for this formula is explained later.

• We can try using this formula by considering this problem.

Worked Example
Retirement Account

Since March of 1960, Mang Jun has been saving his earnings into an account that grows compounding each month for which he deposits at the end of each month. If he is expected to retire in 2025 of the same month, how much will Mang Jun get after his retirement?

Solution

• The problem asks us to know how much will Mang Jun get after retirement.
  • Therefore, this tells us that this is a future value problem.
• We know that the annual interest rate is or
• The money compounds each month and so the compounding frequency is .
• Mang Jun deposits at the end of each month as well, so his payment frequency is also .
• The problem states that the term of the annuity is between and .
  • We can find the number of years between them by subtracting them:

From the given values, we can determine that:

• his periodic interest rate is
• his total number of annuity payments are .

Therefore, the his total retirement savings are:

The Present Value of Annuities